Trigonometry and the Unit Circle Chapter 4
Topics Demonstrate an understanding of angles in standard position, expressed in degrees and radians. Develop and apply the equation of the unit circle. Solve problems, using the six trigonometric ratios for angles expressed in radians and degrees. Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians.
Special Angles Without a calculator determine the following: A) Sin 30 o B) tan 45 o C) sin 60 o
Special Triangles 45 o -45 o -90 o
Special Triangles 30 o -60 o -90 o
From geometry An angle consists of rays that originate from a common point called the vertex. Reflex angle > 180 o 180 o Obtuse angle is between 90 o and 180 o o 90 180 Acute angle is less than 90 o o 0 90 o o
It s Greek To Me! It is customary to use small letters in the Greek alphabet to symbolize angle measurement. alpha beta gamma theta phi delta
To place an angle in standard position, one ray is placed on the postive x-axis with the vertex at the origin. This side is called the initial side. The other side is called the terminal side (or arm). Initial side
Angle Measurements We will be using two different units of measure when talking about angles: Degrees and Radians Let s talk about degrees first. This was covered in Math 00 One degree (1º) is equivalent to a rotation of of one revolution. 1 360
Example: 1. Place the following in standard position: A) 90 o B) 180 o C) 70 o D) 360 o 1 of 360 o 4 1 of 360 o 3 of 360 o 4 o 360 (1 revolution) Note: These are called quadrantal angles These are angles that lie along the x-axis or y-axis
E) 54 o F) 10 o G) 45 o H)-150 o Note: If the angle is positive the terminal arm is swept out from the initial side in a counter-clockwise fashion. A negative angle is swept out in a clockwise fashion It might be helpful to think of a positive rotation as opening upward from standard position, whereas a negative angle opens downward.
Classifying Angles Angles are often classified according to the quadrant in which their terminal sides lie. Example: Name the quadrant in which each angle lies. 50º 08º Quadrant 1 Quadrant 3 II I -75º Quadrant 4 III IV
Definition: Reference Angle: This is the acute angle that is formed by the terminal arm of the angle and either the positive or negative x-axis. Example: Draw the angles in standard position and find the reference angles A)135 o B) 0 o C) -45 o D)-150 o R R R R R
E) 604 o F) 70 o G) -345 o H)-550 o R R R R Note: For angles larger than 360 o or smaller than -360 o we subtract or add multiples of 360 o to determine where the angle is.
Exit Card: Sketch each angle in standard position and find the reference angle. A) 60 o B) -15 o C)-17 o D) 750 o R R R R
E)-330 o F) 4 o G) 70 o h) 1090 o R R R R
Radian Measure A second way to measure angles is in radians. So, what is a radian and why do we use them? (Why not just use degrees?)
What is a radian? A radian is an angle measurement that gives the ratio: length of the arc length of the radius When the length of the arc The angle has a measure of 1 radian = 1 radian equals the length of the radius
The angle has a measure of radians When the length of the arc is twice as long as = radians the length of the radius
When the length of the arc is three times as long as The angle has a measure of 3 radians = 3 radians the length of the radius
What is a radian? A radian is an angle measurement that gives the ratio: length of the arc length of the radius When the length of the arc is π times as long as = π radians The angle has a measure of π radians the length of the radius
What is a radian? A radian is an angle measurement that gives the ratio: length of the arc length of the radius s r s r Arclength = s Radius= r
Why do we use radians? Radians are very important in Calculus. The area between y = sin (x) and the x-axis from 0 < x < 180 is approximately 114.6 when graphed in degrees.
Using radians: The area between y = sin (x) and the x-axis from 0 < x < π is exactly when graphed in radians.
Finding the limits of trigonometric function and subsequently finding derivatives of trig functions work best when using radians In radians: sin lim 1 0 0 In degrees: sin lim 0.01745...
Section 4.1, Figure 4.6, Illustration of Six Radian Lengths, pg. 49 Radian Measure radians corresponds to 360 6.8 radians corresponds to 180 3.14 radians corresponds to 90 1.57 Copyright Houghton Mifflin Company. All rights reserved. Digital Figures, 4 7
Common Radian Measure Section 4.1, Figure 4.7, Common Radian Angles, pg. 49
Conversion of angle measurement 180 º = radians To convert degrees to radians: Degree Measure 180 o Radian Measure Example 1: Convert the following degree measures into radian measures A) 45 o B) 10 o C) 90 o
Ex. Convert the degrees to radian measure. a) 60 b) 30 c) -54 d) -118 e) 5
Conversion of angle measurement To convert radians to degrees: Radian Measure 180 o Degree Measure Example 3: Convert the following radian measures into degree measures 3 5 6 A) B) C) D) 11 4
Ex 4. Convert the radians to degrees. π 6 3 a) b) c) 7 1 11 18 d) e) 9
Degree and Radian Form of Special Angles 90 150 135 10 60 45 30 180 0 360 10 5 40 300 315 330 70
Co-terminal Angles Section 4.1, Figure 4.4, Coterminal Angles, pg. 48 Angles that have the same initial and terminal sides are co-terminal. Angles and are co-terminal. Copyright Houghton Mifflin Company. All rights reserved. Digital Figures, 4 5 34
Finding Co-terminal Angles You can find an angle that is co-terminal to a given angle by adding or subtracting multiples of 360º or Ex 1: Find one positive and one negative angle that are co-terminal to 11º. Give exact answers in reduced form. For a positive co-terminal angle, add 360º : 11º + 360º = 47º For a negative co-terminal angle, subtract 360º: 11º - 360º = -48º
Ex : Find one positive and one negative angle that are co-terminal to 700 o. Give exact answers in reduced form.
Ex 3. Find one positive and one negative angle that π is co-terminal with the angle = in standard position. Give exact answers in reduced form. For a positive co-terminal angle, add 3 For a negative co-terminal angle, subtract Ex 4. Find one positive and one negative angle that is coterminal with the angle = 3 in standard position. Give approximate answers to decimal places
Ex 5. Find all of the angles that are co-terminal with the angle = form. 7 5 from 0 to 6. Give exact answers in reduced
Ex 6. Find all of the angles that are coterminal with the angle = 3.
Radian Measure, Arc Length, and Area Recall the formula for radians: s r Arc length s of a circle is found with the following formula: IMPORTANT: ANGLE s = r MEASURE MUST BE IN RADIANS TO USE FORMULA! arc length radius measure of angle
Ex 1. Find the arc length if we have a circle with a radius of 3 meters and central angle of 0.5 radian. 3 arc length to find is in red = 0.5 s = r
Ex. Find the radius of a circle in which an arc of 3 km subtends a central angle of 0. Remember: If the measure of the angle is in degrees, we can't use the formula until we convert it to radians. s r r s o 0 0 180 3km 7km r 9 9
Ex 3. Given an arc length of 0 cm cut on a circle of radius 5.4 cm, determine the measure of the central angle in radians and degrees.
Fun One! During a family vacation, you go to dinner at the Seattle Space Needle. There is a rotating restaurant at the top of the needle that is circular and has a radius of 40 feet. It makes one rotation per hour. At 6:4 p.m., you take a seat at a window table. You finish dinner at 8:8 p.m. Through what angle did your position rotate during your stay? How many feet did your position revolve?
Text: Page 175-179 # 1,, 4, 6, 7, 8, 9, 11 a) c) e) g), 13, 15c) If the bike tire is 700mm in diameter, how fast is the bike travelling?
Section 4. Your Friend THE UNIT CIRCLE
What does unit circle really mean? It s a circle with a radius of 1 unit centred at the origin, (0, 0). To find the equation of the unit circle we use the Pythagorean Theorem. a b c
1 x y P(x, y) Point P represents any point on the unit circle. Applying the Pythagorean Theorem results in: x x 1 y y 1 This is the equation of the unit circle.
How would the equation differ if the radius was r instead of 1? Thus we can generalize the equation of a circle with centre (0,0) and radius r to be x + y = r.
Example. Determine the equation centred at the origin satisfy the following conditions. A) Radius = 6 B) Radius = 5 C) Radius = D) Diameter = 1 E) Passes through point (5, -1)
Using the unit circle, you should be able to complete the following tasks. Given an angle θ in standard position, expressed in degrees or radians, determine the coordinates of the corresponding point on the unit circle. Conversely, determine an angle in standard position that corresponds to a given point on the unit circle. We start performing these tasks using special angles and then move on to any angles.
Consider the following unit circle, what are the coordinates for the quadrantal angles? Place these values on the blank unit circle
45 o 30 o 60 o EX.
Exact Values Reference Angles of 45 o or 4
Exact Values Reference Angles of 45 o or 4 What are the coordinates? 4, 1, 180 45 0, 0, 360 Now, reflect the triangle to the second quadrant 3
What are the coordinates? -, 3 4 1, 180 45 0, 0, 360-1 4, 3 Now, reflect the triangle to the third quadrant
-, What are the coordinates? 3 4 1, 180 45 0, 0, 360-1 4, -, - 5 4 3 Now, reflect the triangle to the fourth quadrant
-, 3 4 1, 180 45 0, 0, 360-1 4, What are the coordinates? -, - 5 4 3 7 4, - Place these values on the blank unit circle
Reference Angles of 30 o or 6
Now, reflect the triangle to the second quadrant. Reference Angles of 30 o or 6 What are the coordinates? 6 1 1 3 1, 30 3
Now, reflect the triangle to the third quadrant. What are the coordinates? - 3 1, 5 6 1 6 1 1 30 3 1, - 3 3
- 3 1, 5 6 1 6 1 1 30 3 1, What are the coordinates? - 3 1, - 1 7 6-3 3 Now, reflect the triangle to the fourth quadrant.
- 3 1, 5 6 1 1 1 30 6 3 1, - 3 1, - 1 7 6-3 3 1 3 1 11, - 6 What are the coordinates? Place these values on the blank unit circle
Let s look at another family Reference Angles of 60 o or 3
Let s look at another family Reference Angles of 60 o or 3 3 What are the coordinates? 1 3,, 180 60 0, 0, 360 1 1 3 3 Now, reflect the triangle to the second quadrant
What are the coordinates? - 1 3, 3 3 1 3, 3 1 1, 180 60 0, 0, 360 1 1-3 3 Now, reflect the triangle to the third quadrant
- 1 3, 3 3 1 3, What are the coordinates?, 180 60 0, 0, 360 3 3-1 1 1 1 3-1 3, - 4 3 3 Now, reflect the triangle to the fourth quadrant
- 1 3, 3 3 1 3,, 180 60 0, 0, 360 3 3-1 1 1 1 3 3 What are the coordinates? - 1 3, - 4 3 3 5 3 1 3, - Place these values on the blank unit circle
Examples 1. Determine the coordinates of the corresponding points on the unit circle given the following angles in standard position: A) 135 o B) -10 o C) 13 6
. Determine the radian measure for all angles in standard position on the unit circle given the following corresponding points: 1 3, A) (0, 1) B) C),
3. Determine the degree measure for the smallest positive angle in standard position (Principal Angle) on the unit circle given the following corresponding points: 1 1 A) (0, -1) B),
4. Determine the coordinates for all points on the unit circle that satisfy the conditions given. Draw a diagram in each case. A) The x-coordinate is 4 5
4. Determine the coordinates for all points on the unit circle that satisfy the conditions given. Draw a diagram in each case. B) The y-coordinate is 7 5
5. Determine if the points are on the unit circle. If they are not on the unit circle what is the required radius for the points to be on a circle centred at origin? A),3 B) 6, 5 7 7 C) 1 4, 3 4
6. If P() is a point at the intersection of the terminal arm of angle and the unit circle, determine the exact coordinates of each of the following. A) P ( π) B) P( 5π 6 ) C) P(1080o )
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4.3 Trigonometric Ratios
In Mathematics 101 and 00, you worked with the three primary trigonometric ratios. What are they?
In this section you will be introduced to the reciprocal ratios: csc θ, sec θ and cot θ. Also, we explore how we can find trigonometric ratios for angles bigger than 90 o and for negative angles. This will be done by applying the trig ratios to the unit circle.
The Unit Circle xy, sin y 1 y r = 1 y cos x 1 x x tan y x sin cos
Terminal Points A terminal point is the point where the terminal side of the angle intersects the unit circle. Coordinates are (x, y) or (cos, sin )
Reciprocal Ratios P(x, y) 1 y x Note: x + y = 1 Sine fn: Cosine fn Tangent fn Cotangent fn Secant fn Cosecant fn y sin 1 x cos 1 tan y x cot 1 sec x 1 csc y x y Point P (x, y) can be written as P (cos, sin)
Which pairs are reciprocals of each other? 1 x y P(x, y) Sine fn: Cosine fn Tangent fn Cotangent Secant fn Cosecant fn y sin 1 x cos 1 y tan x x cot y 1 sec x 1 csc y 1 tan cot 1 cot tan cos sec 1 sec 1 cos 1 sin csc 1 csc sin
Note: If cot 1 tan and tan y x sin cos then cot cos sin
The Unit Circle sin 0 csc0 cos 0 sec0 tan0 cot 0
The Unit Circle sin csc cos sec tan cot
The Unit Circle sin csc cos sec tan cot
Examples 1.Evaluate the six trigonometric functions at each real number. 3 Where is the terminal arm? 1, 3 What is the reference angle? Sin Cos Tan 3 3 3 = y = x y x 3 1 3 1 What are the coordinates? 3 3 1
3, 1 3 Sin 3 Cos 3 Tan 3 1 3 3 Csc 3 3 3 3 Cot 3 3 3 Sec 1 3 3 3 3 3
. Evaluate the six trigonometric functions at each real number. 7 4 Where is the terminal arm? What is the reference angle?, What are the coordinates? Sin 7 4 Csc 7 4 Cos Tan 7 4 7 4-1 Sec Cot 7 4 7 4-1 So, you think you got it now?
3. Evaluate the six trigonometric functions at each real number. (0, -1) Sin = y = -1 Csc = -1 Cos = x = 0 1 Sec DNE 0 Tan y x 1 DNE 0 0 Cot 1 = 0 Does Not Exist
4.Evaluate the six trigonometric functions at. 0 660
Page 01 #1 Quiz tomorrow NO Calculator Some questions MAY come from homework For Each Question you must do the following: What is the exact value for each trigonometric ratio? Show all necessary work in places provided. (Location of terminal arm, reference angle and coordinates of points)
Approximate Values of Trigonometric Ratios (Using a Calculator)
Calculators can be use to obtain approximate values for sine, cosine and tangent. Most calculators can determine trig values for angles measured in degrees (Deg), and radians (Rad), and even in gradients (Grad)
GRADIENTS (GRADES) rise grad 100% run h = 100% d
Determine the following A) Sin 30 B) Sin 30 o In which quadrant does an angle of 30 terminate?
Using Calculator for finding Exact Values Some students rely on their calculator to find exact values. For example, if a calculation results in 0.7071, these students have memorized that 0.7071 =
If you want to be one of these students you need to know these approximations 1 3 3 3 3 3 3 Remember this only helps if you have access to a scientific calculator.
You can find the values of reciprocal trig functions (csc, sec, cot) using the correct reciprocal relationship. Determine the following, correct to 4 decimal places: A) sec60 B) cot( 60) C) 17 csc 6
D) sec5 48'30"
If the angle is not exactly to the next degree it can be expressed as a decimal (most common in math) or in degrees, minutes and seconds (common in surveying and some navigation). 1 degree = 60 minutes 1 minute = 60 seconds = 5 48'30" degrees seconds minutes To convert to decimal form use conversion fractions. These are fractions where the numerator = denominator but in two different units. Put unit on top you want to convert to and put unit on bottom you want to get rid of. Let's convert the seconds to minutes 30" 1' 60" = 0.5'
1 degree = 60 minutes 1 minute = 60 seconds = 5 48'30" = 5 48.5' = 5.808 Now let's use another conversion fraction to get rid of minutes. 48.5' 1 60' =.808
D) sec5 48'30" sec5.808 1 cos 5.808
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Simplify Trigonometric Expressions From Mathematics 00, you should be familiar with performing operations on rational expressions and expressions involving radicals Example: Simplify x 16 x 4x 5 0
Simplify Trigonometric Expressions In this section the rational expressions will involve exact values of trigonometric functions cos sin 6 o tan 30 After you obtain an answer you could use a calculator to check your solution. The emphasis here, however, is on finding exact values using the unit circle, reference triangles, and mental math strategies.
Examples 6 6 1. sin cos. cos 3 sin 4 7 cos 4 7 6
6 3 4 4 3 sin 3. cos tan sin
4. sin 45 sin10 o o sin10 cos30 o o
5. cos sin 6 o tan30
PRACTICE: Find the exact value of the following expressions: 11 ( i ) sin cos 3 6 11 ( ii ) csc cot 3 4 ( iii )cot ( iv ) 6 7 11 cot cos 3 3 o csc 40 Text Page 0. #9
11 ( i ) sin cos 3 6
11 ( ii ) csc cot 3 4
( iii )cot 6 7
( iv ) 11 cot cos 3 3 o csc 40
Finding angles Determine, with or without technology, the measures, in degrees or radians, of the angles in a specified domain, given the value of a trigonometric ratio. given a point on the terminal arm of an angle in standard position.
Examples of finding angles given the value of a trigonometric ratio. 1. Determine the value of θ when for the domain π θ π. Solution: Determine the reference angle, You could think about a triangle with hypotenuse and adjacent side 1. Alternatively, you could apply the reciprocal ratio 1 cos sec
Once the reference angle is determined, identify the quadrants where the secant ratio is negative. The final step focuses on identifying all possible values within the given domain: π θ π. 5 3 3 5,,, 4 4 4 4
.Determine the value of θ when for the domain 0 θ π. sin 3
3.Determine the value of θ when csc for the domain 0 o θ 360 o
4.Determine the value of θ when for the domain -π θ 4π sec 3 3
Other Examples 5. If the terminal arm of is in the second quadrant and sin, determine sin cos
1 3 6. If the point, is on the terminal arm of, determine A) The quadrant is in B) The principal angle for, in radians Recall: The principal angle is the first positive angle that ends on the terminal arm C) tan
7. The point P(-0.7880, 0.6157) is the image of the point (1, 0) rotated through. Find. Find the reference angle first. Solve cos R = 0.7880 R = 38 o Take the positive value Since is in the nd quadrant = 180 o -38 o = 14 o
New Definitions of the Trigonometric Functions Consider a circle with radius r, centre at the origin. The terminal side of an angle, in standard position intersects the circle at the point P, with coordinates (x, y).
P(x, y) r y x Note: x + y = r Sine fn: Cosine fn Tangent fn Cotangent Secant fn Cosecant fn y sin r x cos r y tan x x cot y r sec x r csc y Point P (x, y) can be written as P (rcos, rsin)
Note: The radius is always positive, but the x and y may be positive or negative depending on what quadrant point P lies in. Ex: Where would be if both x and y are negative?
Examples 1. Find the exact value of csc if and is in the third quadrant. cos 3 8 Bonus: What is in degrees?
Examples. Find the exact value of sin if and is in the fourth quadrant. tan 5 1 Bonus: What is in radians
Example 3: The following points are on the terminal arm of which intersect a circle with centre (0, 0) and radius r. For each point: A) Draw a diagram showing as a principal angle, in standard position. B) Find the radius of the circle. C) Find the exact values of the 6 trig functions D) Determine the reference angle R E) Find
(i) P(3, 4)
(ii) P(-3, 4)
(iii) P(-3, -5)
(iv) P(1, -1)
(v) P(0, -)
Example 4. The point P(-3.7157, -3.3457) is the point of intersection between the terminal arm of and a circle of radius r centred at the origin. A) Find r x + y = r B) Find. Find the reference angle. r ( 3.7157) ( 3.3457) = 5 cos R x r 3.7157 5 R = 4 o = 180 o + 4 o = o
5. A point on a circle with radius 8 rotates at 14.15 revolutions per minute. Find the exact location of the point after 3 minutes, assuming that the dot started at (8, 0).
6. Find the approximate measure of all angles when sec 3.5 in the domain - θ. Give answers to decimal places.
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Introduction to Solving Trigonometric Equations Section 4.4
In Mathematics 00, you solved simple trigonometric equations of the form sin x 0.5 tan x 3.
In Math 300: This will now be extended to include trigonometric equations with all six trigonometric ratios. We will solve first and second degree trigonometric equations with the domain expressed in degrees and radians.
First Degree Equations When solving first degree equations, rearrangement will sometimes be necessary to isolate the trigonometric ratio. Example: 1. Solve cosx 1 0, x,
You should always check all solutions with a calculator or by using the unit circle where appropriate. When solving equations you should also check that the solutions are defined for the domain of the tan, cot, sec and csc functions.
. Find the values of x in degrees o o where secx 1 1, x 0,360
3. Solve: 4 cot + 3 = - cot - 8; 0,
Second Degree Equations We solve second degree equations through techniques such as factoring (e.g., sin θ - 3sinθ + = 0, for all θ) or isolation and square root principles (e.g., tan θ - 3 = 0, for all θ).
4. Solve sin θ - 3sin θ + = 0, for all θ (in degrees). Solution: This is similar to solving x 3x + = 0 (x )(x 1) = 0 So, sin θ - 3sin θ + = 0 factors to: (sin θ )(sin θ 1) = 0 sin θ = sin θ = 1 θ = θ =
If the domain is real numbers, there are an infinite number of rotations on the unit circle in both a positive and negative direction. So to find all θ (in degrees) we write an expression for the values corresponding to θ = 90 o o o 90 360 k,where k
5. Solve: tan θ - 3 = 0, for all θ (in radians). Solution: This can be solved by factoring a difference of squares or by isolation and square root principles tan 3 0 tan 3 tan 3 isolation square root tan 3 0 You should realize that using only the principal square root 3 in this equation causes a loss of roots.
Another common error occurs when you do not find all solutions for the given domain. Remember to focus on the given domain. tan 3 In the equation above, the reference angle is and since there are two cases to consider (tangent being negative and positive), there are solutions in all four quadrants.
6. Solve: cos x 5cos x 0 where 0 θ < π
7. Solve: sin x + 5 sin x - 3 = 0; x ( π, π). Give exact solutions, or round to the nearest one hundredth.
8. Solve: 5 sec x = 1- sec x; for all x in radians. Give exact solutions, or round to the nearest one hundredth.
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